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Digital-to-Digital Frequency Transformations

Module by: Douglas L. Jones. E-mail the author

Given a prototype digital filter design, transformations similar to the bilinear transform can also be developed.

Requirements on such a mapping z-1=g⁢z-1 z -1 g z -1 :

  1. points inside the unit circle stay inside the unit circle (condition to preserve stability)
  2. unit circle is mapped to itself (preserves frequency response)

This condition implies that e−(i⁢ ω 1 )=g⁢e−(i⁢ω)=|g⁢ω|⁢ei⁢∠⁢g⁢ω ω 1 g ω g ω g ω requires that |g⁢e−(i⁢ω)|=1 g ω 1 on the unit circle!

Thus we require an all-pass transformation: g⁢z-1=∏ k =1pz-1− α k 1− α k ⁢z-1 g z -1 k 1 p z -1 α k 1 α k z -1 where | α K |<1 α K 1 , which is required to satisfy this condition.

Example 1: Lowpass-to-Lowpass

z 1 -1=z-1−a1−a⁢z-1 z 1 -1 z -1 a 1 a z -1 which maps original filter with a cutoff at ωc ωc to a new filter with cutoff ωc ′ ωc ′ , a=sin12⁢( ω c − ω c ′ )sin12⁢( ω c + ω c ′ ) a 1 2 ω c ω c ′ 1 2 ω c ω c ′

Example 2: Lowpass-to-Highpass

z 1 -1=z-1+a1+a⁢z-1 z 1 -1 z -1 a 1 a z -1 which maps original filter with a cutoff at ωc ωc to a frequency reversed filter with cutoff ωc ′ ωc ′ , a=cos12⁢( ω c − ω c ′ )cos12⁢( ω c + ω c ′ ) a 1 2 ω c ω c ′ 1 2 ω c ω c ′

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